Problem: Add the following rational expressions. $\dfrac{m-9}{9m+3}+\dfrac{4}{7m^2}=$
Answer: We can add two rational expressions whose denominators are equal by adding the numerators and keeping the denominator the same. [Does this fit with how we add rational numbers?] When the denominators are not the same, we must manipulate them so that they become the same. In other words, we must find a common denominator. Since the two denominators do not share any common factors, the common denominator is simply the product of these two denominators: $({9m+3})\cdot({7m^2})$. Let's manipulate the expressions to have that denominator: $\begin{aligned} &\phantom{=}\dfrac{m-9}{{9m+3}}+\dfrac{4}{{7m^2}} \\\\ &=\dfrac{(m-9)\cdot({7m^2})}{({9m+3})\cdot({7m^2})}+\dfrac{4\cdot({9m+3})}{({7m^2})\cdot({9m+3})} \end{aligned}$ [Why did we do that?] Now that both denominators are the same, let's add! $\begin{aligned} &\phantom{=}\dfrac{(m-9)\cdot(7m^2)}{(9m+3)\cdot(7m^2)}+\dfrac{4\cdot(9m+3)}{(7m^2)\cdot(9m+3)} \\\\ &=\dfrac{(m-9)\cdot(7m^2)+4\cdot(9m+3)}{(9m+3)(7m^2)} \\\\ &=\dfrac{7m^3-63m^2+36m+12}{(9m+3)(7m^2)} \end{aligned}$ In conclusion, $\dfrac{m-9}{9m+3}+\dfrac{4}{7m^2}=\dfrac{7m^3-63m^2+36m+12}{(9m+3)(7m^2)}$